According to the quantum regression theorem [1]}, [2]} the correlation \(\langle \hat{a}_{\bf k}^\dag (t) \hat{a}_{\bf k}(t+\tau )\rangle \)  obeys
\( \frac{d}{d\tau }{\rm tr} \left( \hat{a}_{\bf k} \hat{\rho }(\tau ) \hat{a}_{\bf k}^\dag \right)= {\rm tr}\left( \hat{a}_{\bf k} L_{\rm pump}(\hat{\rho }(\tau ) \hat{a}_{\bf k}^\dag ) \right) \\+ {\rm tr}\left( \hat{a}_{\bf k} L_{\rm diss}(\hat{\rho }(\tau ) \hat{a}_{\bf k}^\dag ) \right)+ {\rm tr}\left( \hat{a}_{\bf k} L_{\rm therm}(\hat{\rho }(\tau ) \hat{a}_{\bf k}^\dag ) \right),\) 
